This model is formed of a pyramid on top of a prism, and its height is 22 inches. Which has a greater surface area, the prism or the pyramid?
To solve this question, we need to break up this composite shape into its two pieces, a prism and a pyramid. Let’s start by finding the surface area of the prism. The surface area of the prism would be equal to two times 𝑎𝑏 plus two times 𝑏𝑐 plus two times 𝑎𝑐 for a prism with side lengths 𝑎, 𝑏, and 𝑐. In the prism that we’re talking about, we have a base of 12 by 12. The base is a square, and then the prism has a height of six inches. We can plug this in to the surface area formula to get two times 12 times 12 plus two times 12 times six plus two times 12 times six. 12 times 12 is 144 and two times that is 288. Six times two is 12 and 12 times 12 equals 144. So we’ll have 288 plus 144 plus 144. When we add this all together, we get 576. This is a sum of all the outside areas of the prism. And that means the units will be inches squared.
But now, we need to move on and think about the surface area of the pyramid. This pyramid has five faces. We know that we will take the area of the base and we’ll need to add the area of the four triangular faces. Because the base of this pyramid is a square, we can find the area of one of the triangular faces and multiply that by four since all four faces are equal. The base of this pyramid again, as we’ve already said, is a square. And the square has a side of 12. 12 squared will give us the area of the base. This means our main goal now is to find the area of the triangular face on the outside of this pyramid.
To find the area of a triangle, we take one-half its height times its base. The base of this triangle is equal to 12 inches. And the height is here. But finding this height is a little bit more complicated. We call this the slant height of the prism. But to find that, we’ll need to consider the right triangle created with the height of the whole pyramid. We know that the height of the pyramid on top of the prism is 22 inches. And so, the height of the pyramid will be whatever we add to six to get 22. 16 inches plus six inches equals 22 inches. And so, we can say that the height of the pyramid is 16 inches.
Because the height creates a perpendicular angle with the base, we know that the distance from the height to the edge of the pyramid is six inches. It’s half of 12. And because this is a right triangle created here, we can use the Pythagorean theorem to find the slant height. The square of the height we’re looking for is equal to 16 squared plus six squared. ℎ squared is equal to 292. And so, we take the square root of both sides. And instead of simplifying, we can just leave this as ℎ is equal to the square root of 292. That’s a measure of inches. And then, we can plug the square root of 292 in for the height. And we’ve already said the base is equal to 12.
The surface area of this pyramid is going to be equal to 144 plus four times one-half times the square root of 292 times 12 which is equal to 544.11217 continuing. Again, as this is a measure of area, the units will be inches squared. Our job was to compare the surface areas of the two shapes. We know that 576 is greater than 544.112 continuing. 576 inches squared was the surface area of the prism. And so, we can say that the prism has a larger surface area than the pyramid.